LMFDB - Embedded newform 462.4.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [462,4,Mod(1,462)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(462, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("462.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 462 = 2 \cdot 3 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	462.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$27.2588824227$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	462.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -13.0000 q^{5} +6.00000 q^{6} -7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -13.0000 q^{5} +6.00000 q^{6} -7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} -26.0000 q^{10} +11.0000 q^{11} +12.0000 q^{12} -67.0000 q^{13} -14.0000 q^{14} -39.0000 q^{15} +16.0000 q^{16} +8.00000 q^{17} +18.0000 q^{18} +21.0000 q^{19} -52.0000 q^{20} -21.0000 q^{21} +22.0000 q^{22} -194.000 q^{23} +24.0000 q^{24} +44.0000 q^{25} -134.000 q^{26} +27.0000 q^{27} -28.0000 q^{28} -221.000 q^{29} -78.0000 q^{30} +88.0000 q^{31} +32.0000 q^{32} +33.0000 q^{33} +16.0000 q^{34} +91.0000 q^{35} +36.0000 q^{36} -347.000 q^{37} +42.0000 q^{38} -201.000 q^{39} -104.000 q^{40} +292.000 q^{41} -42.0000 q^{42} -458.000 q^{43} +44.0000 q^{44} -117.000 q^{45} -388.000 q^{46} +221.000 q^{47} +48.0000 q^{48} +49.0000 q^{49} +88.0000 q^{50} +24.0000 q^{51} -268.000 q^{52} -642.000 q^{53} +54.0000 q^{54} -143.000 q^{55} -56.0000 q^{56} +63.0000 q^{57} -442.000 q^{58} +273.000 q^{59} -156.000 q^{60} -530.000 q^{61} +176.000 q^{62} -63.0000 q^{63} +64.0000 q^{64} +871.000 q^{65} +66.0000 q^{66} +561.000 q^{67} +32.0000 q^{68} -582.000 q^{69} +182.000 q^{70} +604.000 q^{71} +72.0000 q^{72} +703.000 q^{73} -694.000 q^{74} +132.000 q^{75} +84.0000 q^{76} -77.0000 q^{77} -402.000 q^{78} +552.000 q^{79} -208.000 q^{80} +81.0000 q^{81} +584.000 q^{82} -144.000 q^{83} -84.0000 q^{84} -104.000 q^{85} -916.000 q^{86} -663.000 q^{87} +88.0000 q^{88} +750.000 q^{89} -234.000 q^{90} +469.000 q^{91} -776.000 q^{92} +264.000 q^{93} +442.000 q^{94} -273.000 q^{95} +96.0000 q^{96} -1370.00 q^{97} +98.0000 q^{98} +99.0000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	4.00000	0.500000
$5$	−13.0000	−1.16276	−0.581378	−	0.813634i	$-0.697486\pi$
$5$	−13.0000	−1.16276	−0.581378	+	0.813634i	$0.697486\pi$
$6$	6.00000	0.408248
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	−26.0000	−0.822192
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	12.0000	0.288675
$13$	−67.0000	−1.42942	−0.714710	−	0.699421i	$-0.753441\pi$
$13$	−67.0000	−1.42942	−0.714710	+	0.699421i	$0.753441\pi$
$14$	−14.0000	−0.267261
$15$	−39.0000	−0.671317
$16$	16.0000	0.250000
$17$	8.00000	0.114134	0.0570672	−	0.998370i	$-0.481825\pi$
$17$	8.00000	0.114134	0.0570672	+	0.998370i	$0.481825\pi$
$18$	18.0000	0.235702
$19$	21.0000	0.253565	0.126782	−	0.991931i	$-0.459535\pi$
$19$	21.0000	0.253565	0.126782	+	0.991931i	$0.459535\pi$
$20$	−52.0000	−0.581378
$21$	−21.0000	−0.218218
$22$	22.0000	0.213201
$23$	−194.000	−1.75877	−0.879387	−	0.476108i	$-0.842047\pi$
$23$	−194.000	−1.75877	−0.879387	+	0.476108i	$0.842047\pi$
$24$	24.0000	0.204124
$25$	44.0000	0.352000
$26$	−134.000	−1.01075
$27$	27.0000	0.192450
$28$	−28.0000	−0.188982
$29$	−221.000	−1.41513	−0.707563	−	0.706650i	$-0.750206\pi$
$29$	−221.000	−1.41513	−0.707563	+	0.706650i	$0.750206\pi$
$30$	−78.0000	−0.474693
$31$	88.0000	0.509847	0.254924	−	0.966961i	$-0.417950\pi$
$31$	88.0000	0.509847	0.254924	+	0.966961i	$0.417950\pi$
$32$	32.0000	0.176777
$33$	33.0000	0.174078
$34$	16.0000	0.0807052
$35$	91.0000	0.439480
$36$	36.0000	0.166667
$37$	−347.000	−1.54180	−0.770898	−	0.636959i	$-0.780192\pi$
$37$	−347.000	−1.54180	−0.770898	+	0.636959i	$0.780192\pi$
$38$	42.0000	0.179297
$39$	−201.000	−0.825276
$40$	−104.000	−0.411096
$41$	292.000	1.11226	0.556131	−	0.831095i	$-0.312285\pi$
$41$	292.000	1.11226	0.556131	+	0.831095i	$0.312285\pi$
$42$	−42.0000	−0.154303
$43$	−458.000	−1.62429	−0.812144	−	0.583458i	$-0.801699\pi$
$43$	−458.000	−1.62429	−0.812144	+	0.583458i	$0.801699\pi$
$44$	44.0000	0.150756
$45$	−117.000	−0.387585
$46$	−388.000	−1.24364
$47$	221.000	0.685876	0.342938	−	0.939358i	$-0.388578\pi$
$47$	221.000	0.685876	0.342938	+	0.939358i	$0.388578\pi$
$48$	48.0000	0.144338
$49$	49.0000	0.142857
$50$	88.0000	0.248902
$51$	24.0000	0.0658955
$52$	−268.000	−0.714710
$53$	−642.000	−1.66388	−0.831939	−	0.554868i	$-0.812769\pi$
$53$	−642.000	−1.66388	−0.831939	+	0.554868i	$0.812769\pi$
$54$	54.0000	0.136083
$55$	−143.000	−0.350584
$56$	−56.0000	−0.133631
$57$	63.0000	0.146396
$58$	−442.000	−1.00065
$59$	273.000	0.602400	0.301200	−	0.953561i	$-0.402613\pi$
$59$	273.000	0.602400	0.301200	+	0.953561i	$0.402613\pi$
$60$	−156.000	−0.335659
$61$	−530.000	−1.11245	−0.556226	−	0.831031i	$-0.687751\pi$
$61$	−530.000	−1.11245	−0.556226	+	0.831031i	$0.687751\pi$
$62$	176.000	0.360516
$63$	−63.0000	−0.125988
$64$	64.0000	0.125000
$65$	871.000	1.66207
$66$	66.0000	0.123091
$67$	561.000	1.02294	0.511471	−	0.859301i	$-0.329101\pi$
$67$	561.000	1.02294	0.511471	+	0.859301i	$0.329101\pi$
$68$	32.0000	0.0570672
$69$	−582.000	−1.01543
$70$	182.000	0.310759
$71$	604.000	1.00960	0.504800	−	0.863236i	$-0.331566\pi$
$71$	604.000	1.00960	0.504800	+	0.863236i	$0.331566\pi$
$72$	72.0000	0.117851
$73$	703.000	1.12712	0.563561	−	0.826074i	$-0.309431\pi$
$73$	703.000	1.12712	0.563561	+	0.826074i	$0.309431\pi$
$74$	−694.000	−1.09021
$75$	132.000	0.203227
$76$	84.0000	0.126782
$77$	−77.0000	−0.113961
$78$	−402.000	−0.583558
$79$	552.000	0.786137	0.393069	−	0.919509i	$-0.371413\pi$
$79$	552.000	0.786137	0.393069	+	0.919509i	$0.371413\pi$
$80$	−208.000	−0.290689
$81$	81.0000	0.111111
$82$	584.000	0.786488
$83$	−144.000	−0.190434	−0.0952172	−	0.995457i	$-0.530355\pi$
$83$	−144.000	−0.190434	−0.0952172	+	0.995457i	$0.530355\pi$
$84$	−84.0000	−0.109109
$85$	−104.000	−0.132710
$86$	−916.000	−1.14854
$87$	−663.000	−0.817024
$88$	88.0000	0.106600
$89$	750.000	0.893257	0.446628	−	0.894720i	$-0.352625\pi$
$89$	750.000	0.893257	0.446628	+	0.894720i	$0.352625\pi$
$90$	−234.000	−0.274064
$91$	469.000	0.540270
$92$	−776.000	−0.879387
$93$	264.000	0.294360
$94$	442.000	0.484987
$95$	−273.000	−0.294834
$96$	96.0000	0.102062
$97$	−1370.00	−1.43405	−0.717023	−	0.697050i	$-0.754496\pi$
$97$	−1370.00	−1.43405	−0.717023	+	0.697050i	$0.754496\pi$
$98$	98.0000	0.101015
$99$	99.0000	0.100504
$100$	176.000	0.176000
$101$	1980.00	1.95067	0.975333	−	0.220736i	$-0.0708461\pi$
$101$	1980.00	1.95067	0.975333	+	0.220736i	$0.0708461\pi$
$102$	48.0000	0.0465952
$103$	−666.000	−0.637116	−0.318558	−	0.947903i	$-0.603199\pi$
$103$	−666.000	−0.637116	−0.318558	+	0.947903i	$0.603199\pi$
$104$	−536.000	−0.505376
$105$	273.000	0.253734
$106$	−1284.00	−1.17654
$107$	1439.00	1.30013	0.650063	−	0.759881i	$-0.274743\pi$
$107$	1439.00	1.30013	0.650063	+	0.759881i	$0.274743\pi$
$108$	108.000	0.0962250
$109$	−956.000	−0.840075	−0.420038	−	0.907507i	$-0.637983\pi$
$109$	−956.000	−0.840075	−0.420038	+	0.907507i	$0.637983\pi$
$110$	−286.000	−0.247900
$111$	−1041.00	−0.890156
$112$	−112.000	−0.0944911
$113$	−2302.00	−1.91641	−0.958203	−	0.286088i	$-0.907645\pi$
$113$	−2302.00	−1.91641	−0.958203	+	0.286088i	$0.907645\pi$
$114$	126.000	0.103517
$115$	2522.00	2.04502
$116$	−884.000	−0.707563
$117$	−603.000	−0.476473
$118$	546.000	0.425961
$119$	−56.0000	−0.0431388
$120$	−312.000	−0.237346
$121$	121.000	0.0909091
$122$	−1060.00	−0.786622
$123$	876.000	0.642165
$124$	352.000	0.254924
$125$	1053.00	0.753465
$126$	−126.000	−0.0890871
$127$	608.000	0.424813	0.212407	−	0.977181i	$-0.431870\pi$
$127$	608.000	0.424813	0.212407	+	0.977181i	$0.431870\pi$
$128$	128.000	0.0883883
$129$	−1374.00	−0.937783
$130$	1742.00	1.17526
$131$	−192.000	−0.128054	−0.0640272	−	0.997948i	$-0.520394\pi$
$131$	−192.000	−0.128054	−0.0640272	+	0.997948i	$0.520394\pi$
$132$	132.000	0.0870388
$133$	−147.000	−0.0958385
$134$	1122.00	0.723329
$135$	−351.000	−0.223772
$136$	64.0000	0.0403526
$137$	−1212.00	−0.755826	−0.377913	−	0.925841i	$-0.623358\pi$
$137$	−1212.00	−0.755826	−0.377913	+	0.925841i	$0.623358\pi$
$138$	−1164.00	−0.718016
$139$	−1876.00	−1.14475	−0.572375	−	0.819992i	$-0.693978\pi$
$139$	−1876.00	−1.14475	−0.572375	+	0.819992i	$0.693978\pi$
$140$	364.000	0.219740
$141$	663.000	0.395991
$142$	1208.00	0.713895
$143$	−737.000	−0.430986
$144$	144.000	0.0833333
$145$	2873.00	1.64545
$146$	1406.00	0.796996
$147$	147.000	0.0824786
$148$	−1388.00	−0.770898
$149$	−265.000	−0.145702	−0.0728512	−	0.997343i	$-0.523210\pi$
$149$	−265.000	−0.145702	−0.0728512	+	0.997343i	$0.523210\pi$
$150$	264.000	0.143703
$151$	−1728.00	−0.931276	−0.465638	−	0.884975i	$-0.654175\pi$
$151$	−1728.00	−0.931276	−0.465638	+	0.884975i	$0.654175\pi$
$152$	168.000	0.0896487
$153$	72.0000	0.0380448
$154$	−154.000	−0.0805823
$155$	−1144.00	−0.592828
$156$	−804.000	−0.412638
$157$	−884.000	−0.449369	−0.224684	−	0.974432i	$-0.572135\pi$
$157$	−884.000	−0.449369	−0.224684	+	0.974432i	$0.572135\pi$
$158$	1104.00	0.555883
$159$	−1926.00	−0.960640
$160$	−416.000	−0.205548
$161$	1358.00	0.664754
$162$	162.000	0.0785674
$163$	673.000	0.323395	0.161698	−	0.986840i	$-0.448303\pi$
$163$	673.000	0.323395	0.161698	+	0.986840i	$0.448303\pi$
$164$	1168.00	0.556131
$165$	−429.000	−0.202410
$166$	−288.000	−0.134657
$167$	1846.00	0.855376	0.427688	−	0.903926i	$-0.359328\pi$
$167$	1846.00	0.855376	0.427688	+	0.903926i	$0.359328\pi$
$168$	−168.000	−0.0771517
$169$	2292.00	1.04324
$170$	−208.000	−0.0938404
$171$	189.000	0.0845216
$172$	−1832.00	−0.812144
$173$	−2816.00	−1.23755	−0.618776	−	0.785567i	$-0.712371\pi$
$173$	−2816.00	−1.23755	−0.618776	+	0.785567i	$0.712371\pi$
$174$	−1326.00	−0.577723
$175$	−308.000	−0.133043
$176$	176.000	0.0753778
$177$	819.000	0.347796
$178$	1500.00	0.631628
$179$	2800.00	1.16917	0.584586	−	0.811332i	$-0.301257\pi$
$179$	2800.00	1.16917	0.584586	+	0.811332i	$0.301257\pi$
$180$	−468.000	−0.193793
$181$	2042.00	0.838567	0.419284	−	0.907855i	$-0.362281\pi$
$181$	2042.00	0.838567	0.419284	+	0.907855i	$0.362281\pi$
$182$	938.000	0.382028
$183$	−1590.00	−0.642274
$184$	−1552.00	−0.621820
$185$	4511.00	1.79273
$186$	528.000	0.208144
$187$	88.0000	0.0344128
$188$	884.000	0.342938
$189$	−189.000	−0.0727393
$190$	−546.000	−0.208479
$191$	3992.00	1.51231	0.756154	−	0.654393i	$-0.227076\pi$
$191$	3992.00	1.51231	0.756154	+	0.654393i	$0.227076\pi$
$192$	192.000	0.0721688
$193$	1526.00	0.569139	0.284570	−	0.958655i	$-0.408149\pi$
$193$	1526.00	0.569139	0.284570	+	0.958655i	$0.408149\pi$
$194$	−2740.00	−1.01402
$195$	2613.00	0.959594
$196$	196.000	0.0714286
$197$	354.000	0.128028	0.0640138	−	0.997949i	$-0.479610\pi$
$197$	354.000	0.128028	0.0640138	+	0.997949i	$0.479610\pi$
$198$	198.000	0.0710669
$199$	1474.00	0.525071	0.262535	−	0.964922i	$-0.415441\pi$
$199$	1474.00	0.525071	0.262535	+	0.964922i	$0.415441\pi$
$200$	352.000	0.124451
$201$	1683.00	0.590595
$202$	3960.00	1.37933
$203$	1547.00	0.534868
$204$	96.0000	0.0329478
$205$	−3796.00	−1.29329
$206$	−1332.00	−0.450509
$207$	−1746.00	−0.586258
$208$	−1072.00	−0.357355
$209$	231.000	0.0764527
$210$	546.000	0.179417
$211$	−590.000	−0.192499	−0.0962495	−	0.995357i	$-0.530685\pi$
$211$	−590.000	−0.192499	−0.0962495	+	0.995357i	$0.530685\pi$
$212$	−2568.00	−0.831939
$213$	1812.00	0.582893
$214$	2878.00	0.919327
$215$	5954.00	1.88865
$216$	216.000	0.0680414
$217$	−616.000	−0.192704
$218$	−1912.00	−0.594023
$219$	2109.00	0.650744
$220$	−572.000	−0.175292
$221$	−536.000	−0.163146
$222$	−2082.00	−0.629436
$223$	−2714.00	−0.814991	−0.407495	−	0.913207i	$-0.633598\pi$
$223$	−2714.00	−0.814991	−0.407495	+	0.913207i	$0.633598\pi$
$224$	−224.000	−0.0668153
$225$	396.000	0.117333
$226$	−4604.00	−1.35510
$227$	−3326.00	−0.972486	−0.486243	−	0.873824i	$-0.661633\pi$
$227$	−3326.00	−0.972486	−0.486243	+	0.873824i	$0.661633\pi$
$228$	252.000	0.0731978
$229$	−1294.00	−0.373406	−0.186703	−	0.982416i	$-0.559780\pi$
$229$	−1294.00	−0.373406	−0.186703	+	0.982416i	$0.559780\pi$
$230$	5044.00	1.44605
$231$	−231.000	−0.0657952
$232$	−1768.00	−0.500323
$233$	3390.00	0.953160	0.476580	−	0.879131i	$-0.341876\pi$
$233$	3390.00	0.953160	0.476580	+	0.879131i	$0.341876\pi$
$234$	−1206.00	−0.336917
$235$	−2873.00	−0.797506
$236$	1092.00	0.301200
$237$	1656.00	0.453877
$238$	−112.000	−0.0305037
$239$	5209.00	1.40980	0.704900	−	0.709307i	$-0.250992\pi$
$239$	5209.00	1.40980	0.704900	+	0.709307i	$0.250992\pi$
$240$	−624.000	−0.167829
$241$	−2511.00	−0.671152	−0.335576	−	0.942013i	$-0.608931\pi$
$241$	−2511.00	−0.671152	−0.335576	+	0.942013i	$0.608931\pi$
$242$	242.000	0.0642824
$243$	243.000	0.0641500
$244$	−2120.00	−0.556226
$245$	−637.000	−0.166108
$246$	1752.00	0.454079
$247$	−1407.00	−0.362450
$248$	704.000	0.180258
$249$	−432.000	−0.109947
$250$	2106.00	0.532781
$251$	4215.00	1.05995	0.529977	−	0.848012i	$-0.322201\pi$
$251$	4215.00	1.05995	0.529977	+	0.848012i	$0.322201\pi$
$252$	−252.000	−0.0629941
$253$	−2134.00	−0.530290
$254$	1216.00	0.300388
$255$	−312.000	−0.0766204
$256$	256.000	0.0625000
$257$	−6945.00	−1.68567	−0.842835	−	0.538172i	$-0.819115\pi$
$257$	−6945.00	−1.68567	−0.842835	+	0.538172i	$0.819115\pi$
$258$	−2748.00	−0.663112
$259$	2429.00	0.582744
$260$	3484.00	0.831033
$261$	−1989.00	−0.471709
$262$	−384.000	−0.0905481
$263$	−57.0000	−0.0133641	−0.00668207	−	0.999978i	$-0.502127\pi$
$263$	−57.0000	−0.0133641	−0.00668207	+	0.999978i	$0.502127\pi$
$264$	264.000	0.0615457
$265$	8346.00	1.93468
$266$	−294.000	−0.0677680
$267$	2250.00	0.515722
$268$	2244.00	0.511471
$269$	2502.00	0.567099	0.283550	−	0.958958i	$-0.408488\pi$
$269$	2502.00	0.567099	0.283550	+	0.958958i	$0.408488\pi$
$270$	−702.000	−0.158231
$271$	565.000	0.126647	0.0633234	−	0.997993i	$-0.479830\pi$
$271$	565.000	0.126647	0.0633234	+	0.997993i	$0.479830\pi$
$272$	128.000	0.0285336
$273$	1407.00	0.311925
$274$	−2424.00	−0.534450
$275$	484.000	0.106132
$276$	−2328.00	−0.507714
$277$	−7164.00	−1.55395	−0.776973	−	0.629534i	$-0.783246\pi$
$277$	−7164.00	−1.55395	−0.776973	+	0.629534i	$0.783246\pi$
$278$	−3752.00	−0.809460
$279$	792.000	0.169949
$280$	728.000	0.155380
$281$	−6985.00	−1.48288	−0.741442	−	0.671017i	$-0.765858\pi$
$281$	−6985.00	−1.48288	−0.741442	+	0.671017i	$0.765858\pi$
$282$	1326.00	0.280008
$283$	−6435.00	−1.35166	−0.675832	−	0.737055i	$-0.736216\pi$
$283$	−6435.00	−1.35166	−0.675832	+	0.737055i	$0.736216\pi$
$284$	2416.00	0.504800
$285$	−819.000	−0.170222
$286$	−1474.00	−0.304753
$287$	−2044.00	−0.420395
$288$	288.000	0.0589256
$289$	−4849.00	−0.986973
$290$	5746.00	1.16351
$291$	−4110.00	−0.827947
$292$	2812.00	0.563561
$293$	8400.00	1.67486	0.837429	−	0.546547i	$-0.184058\pi$
$293$	8400.00	1.67486	0.837429	+	0.546547i	$0.184058\pi$
$294$	294.000	0.0583212
$295$	−3549.00	−0.700443
$296$	−2776.00	−0.545107
$297$	297.000	0.0580259
$298$	−530.000	−0.103027
$299$	12998.0	2.51403
$300$	528.000	0.101614
$301$	3206.00	0.613923
$302$	−3456.00	−0.658511
$303$	5940.00	1.12622
$304$	336.000	0.0633912
$305$	6890.00	1.29351
$306$	144.000	0.0269017
$307$	−6364.00	−1.18310	−0.591552	−	0.806267i	$-0.701484\pi$
$307$	−6364.00	−1.18310	−0.591552	+	0.806267i	$0.701484\pi$
$308$	−308.000	−0.0569803
$309$	−1998.00	−0.367839
$310$	−2288.00	−0.419192
$311$	−7456.00	−1.35946	−0.679728	−	0.733464i	$-0.737902\pi$
$311$	−7456.00	−1.35946	−0.679728	+	0.733464i	$0.737902\pi$
$312$	−1608.00	−0.291779
$313$	10204.0	1.84270	0.921349	−	0.388738i	$-0.127089\pi$
$313$	10204.0	1.84270	0.921349	+	0.388738i	$0.127089\pi$
$314$	−1768.00	−0.317752
$315$	819.000	0.146493
$316$	2208.00	0.393069
$317$	9666.00	1.71261	0.856304	−	0.516472i	$-0.172755\pi$
$317$	9666.00	1.71261	0.856304	+	0.516472i	$0.172755\pi$
$318$	−3852.00	−0.679275
$319$	−2431.00	−0.426677
$320$	−832.000	−0.145344
$321$	4317.00	0.750628
$322$	2716.00	0.470052
$323$	168.000	0.0289405
$324$	324.000	0.0555556
$325$	−2948.00	−0.503156
$326$	1346.00	0.228675
$327$	−2868.00	−0.485018
$328$	2336.00	0.393244
$329$	−1547.00	−0.259237
$330$	−858.000	−0.143125
$331$	−11228.0	−1.86449	−0.932246	−	0.361826i	$-0.882154\pi$
$331$	−11228.0	−1.86449	−0.932246	+	0.361826i	$0.882154\pi$
$332$	−576.000	−0.0952172
$333$	−3123.00	−0.513932
$334$	3692.00	0.604842
$335$	−7293.00	−1.18943
$336$	−336.000	−0.0545545
$337$	2274.00	0.367575	0.183787	−	0.982966i	$-0.441164\pi$
$337$	2274.00	0.367575	0.183787	+	0.982966i	$0.441164\pi$
$338$	4584.00	0.737683
$339$	−6906.00	−1.10644
$340$	−416.000	−0.0663552
$341$	968.000	0.153725
$342$	378.000	0.0597658
$343$	−343.000	−0.0539949
$344$	−3664.00	−0.574272
$345$	7566.00	1.18069
$346$	−5632.00	−0.875081
$347$	−2008.00	−0.310649	−0.155324	−	0.987864i	$-0.549642\pi$
$347$	−2008.00	−0.310649	−0.155324	+	0.987864i	$0.549642\pi$
$348$	−2652.00	−0.408512
$349$	1721.00	0.263963	0.131981	−	0.991252i	$-0.457866\pi$
$349$	1721.00	0.263963	0.131981	+	0.991252i	$0.457866\pi$
$350$	−616.000	−0.0940760
$351$	−1809.00	−0.275092
$352$	352.000	0.0533002
$353$	−2055.00	−0.309849	−0.154924	−	0.987926i	$-0.549513\pi$
$353$	−2055.00	−0.309849	−0.154924	+	0.987926i	$0.549513\pi$
$354$	1638.00	0.245929
$355$	−7852.00	−1.17392
$356$	3000.00	0.446628
$357$	−168.000	−0.0249062
$358$	5600.00	0.826730
$359$	1304.00	0.191706	0.0958530	−	0.995395i	$-0.469442\pi$
$359$	1304.00	0.191706	0.0958530	+	0.995395i	$0.469442\pi$
$360$	−936.000	−0.137032
$361$	−6418.00	−0.935705
$362$	4084.00	0.592957
$363$	363.000	0.0524864
$364$	1876.00	0.270135
$365$	−9139.00	−1.31057
$366$	−3180.00	−0.454156
$367$	−1754.00	−0.249477	−0.124738	−	0.992190i	$-0.539809\pi$
$367$	−1754.00	−0.249477	−0.124738	+	0.992190i	$0.539809\pi$
$368$	−3104.00	−0.439693
$369$	2628.00	0.370754
$370$	9022.00	1.26765
$371$	4494.00	0.628886
$372$	1056.00	0.147180
$373$	11348.0	1.57527	0.787637	−	0.616140i	$-0.211304\pi$
$373$	11348.0	1.57527	0.787637	+	0.616140i	$0.211304\pi$
$374$	176.000	0.0243335
$375$	3159.00	0.435013
$376$	1768.00	0.242494
$377$	14807.0	2.02281
$378$	−378.000	−0.0514344
$379$	−8945.00	−1.21233	−0.606166	−	0.795338i	$-0.707293\pi$
$379$	−8945.00	−1.21233	−0.606166	+	0.795338i	$0.707293\pi$
$380$	−1092.00	−0.147417
$381$	1824.00	0.245266
$382$	7984.00	1.06936
$383$	−5560.00	−0.741783	−0.370891	−	0.928676i	$-0.620948\pi$
$383$	−5560.00	−0.741783	−0.370891	+	0.928676i	$0.620948\pi$
$384$	384.000	0.0510310
$385$	1001.00	0.132508
$386$	3052.00	0.402442
$387$	−4122.00	−0.541429
$388$	−5480.00	−0.717023
$389$	−12372.0	−1.61256	−0.806279	−	0.591535i	$-0.798522\pi$
$389$	−12372.0	−1.61256	−0.806279	+	0.591535i	$0.798522\pi$
$390$	5226.00	0.678535
$391$	−1552.00	−0.200737
$392$	392.000	0.0505076
$393$	−576.000	−0.0739322
$394$	708.000	0.0905293
$395$	−7176.00	−0.914085
$396$	396.000	0.0502519
$397$	−3984.00	−0.503655	−0.251828	−	0.967772i	$-0.581032\pi$
$397$	−3984.00	−0.503655	−0.251828	+	0.967772i	$0.581032\pi$
$398$	2948.00	0.371281
$399$	−441.000	−0.0553324
$400$	704.000	0.0880000
$401$	−414.000	−0.0515565	−0.0257783	−	0.999668i	$-0.508206\pi$
$401$	−414.000	−0.0515565	−0.0257783	+	0.999668i	$0.508206\pi$
$402$	3366.00	0.417614
$403$	−5896.00	−0.728786
$404$	7920.00	0.975333
$405$	−1053.00	−0.129195
$406$	3094.00	0.378208
$407$	−3817.00	−0.464869
$408$	192.000	0.0232976
$409$	4534.00	0.548146	0.274073	−	0.961709i	$-0.411629\pi$
$409$	4534.00	0.548146	0.274073	+	0.961709i	$0.411629\pi$
$410$	−7592.00	−0.914493
$411$	−3636.00	−0.436376
$412$	−2664.00	−0.318558
$413$	−1911.00	−0.227686
$414$	−3492.00	−0.414547
$415$	1872.00	0.221429
$416$	−2144.00	−0.252688
$417$	−5628.00	−0.660922
$418$	462.000	0.0540602
$419$	−5285.00	−0.616203	−0.308102	−	0.951353i	$-0.599694\pi$
$419$	−5285.00	−0.616203	−0.308102	+	0.951353i	$0.599694\pi$
$420$	1092.00	0.126867
$421$	9019.00	1.04408	0.522042	−	0.852920i	$-0.325171\pi$
$421$	9019.00	1.04408	0.522042	+	0.852920i	$0.325171\pi$
$422$	−1180.00	−0.136117
$423$	1989.00	0.228625
$424$	−5136.00	−0.588269
$425$	352.000	0.0401753
$426$	3624.00	0.412168
$427$	3710.00	0.420467
$428$	5756.00	0.650063
$429$	−2211.00	−0.248830
$430$	11908.0	1.33548
$431$	4447.00	0.496994	0.248497	−	0.968633i	$-0.420063\pi$
$431$	4447.00	0.496994	0.248497	+	0.968633i	$0.420063\pi$
$432$	432.000	0.0481125
$433$	−3508.00	−0.389339	−0.194669	−	0.980869i	$-0.562363\pi$
$433$	−3508.00	−0.389339	−0.194669	+	0.980869i	$0.562363\pi$
$434$	−1232.00	−0.136262
$435$	8619.00	0.949999
$436$	−3824.00	−0.420038
$437$	−4074.00	−0.445963
$438$	4218.00	0.460146
$439$	−13187.0	−1.43367	−0.716835	−	0.697243i	$-0.754410\pi$
$439$	−13187.0	−1.43367	−0.716835	+	0.697243i	$0.754410\pi$
$440$	−1144.00	−0.123950
$441$	441.000	0.0476190
$442$	−1072.00	−0.115362
$443$	−8016.00	−0.859710	−0.429855	−	0.902898i	$-0.641435\pi$
$443$	−8016.00	−0.859710	−0.429855	+	0.902898i	$0.641435\pi$
$444$	−4164.00	−0.445078
$445$	−9750.00	−1.03864
$446$	−5428.00	−0.576285
$447$	−795.000	−0.0841213
$448$	−448.000	−0.0472456
$449$	−5724.00	−0.601631	−0.300815	−	0.953682i	$-0.597259\pi$
$449$	−5724.00	−0.601631	−0.300815	+	0.953682i	$0.597259\pi$
$450$	792.000	0.0829672
$451$	3212.00	0.335360
$452$	−9208.00	−0.958203
$453$	−5184.00	−0.537672
$454$	−6652.00	−0.687652
$455$	−6097.00	−0.628202
$456$	504.000	0.0517587
$457$	2306.00	0.236040	0.118020	−	0.993011i	$-0.462345\pi$
$457$	2306.00	0.236040	0.118020	+	0.993011i	$0.462345\pi$
$458$	−2588.00	−0.264038
$459$	216.000	0.0219652
$460$	10088.0	1.02251
$461$	−16068.0	−1.62334	−0.811672	−	0.584114i	$-0.801442\pi$
$461$	−16068.0	−1.62334	−0.811672	+	0.584114i	$0.801442\pi$
$462$	−462.000	−0.0465242
$463$	−16187.0	−1.62478	−0.812391	−	0.583114i	$-0.801834\pi$
$463$	−16187.0	−1.62478	−0.812391	+	0.583114i	$0.801834\pi$
$464$	−3536.00	−0.353782
$465$	−3432.00	−0.342269
$466$	6780.00	0.673986
$467$	−13503.0	−1.33800	−0.668998	−	0.743264i	$-0.733277\pi$
$467$	−13503.0	−1.33800	−0.668998	+	0.743264i	$0.733277\pi$
$468$	−2412.00	−0.238237
$469$	−3927.00	−0.386635
$470$	−5746.00	−0.563922
$471$	−2652.00	−0.259443
$472$	2184.00	0.212980
$473$	−5038.00	−0.489741
$474$	3312.00	0.320939
$475$	924.000	0.0892548
$476$	−224.000	−0.0215694
$477$	−5778.00	−0.554626
$478$	10418.0	0.996879
$479$	−1588.00	−0.151477	−0.0757386	−	0.997128i	$-0.524131\pi$
$479$	−1588.00	−0.151477	−0.0757386	+	0.997128i	$0.524131\pi$
$480$	−1248.00	−0.118673
$481$	23249.0	2.20387
$482$	−5022.00	−0.474576
$483$	4074.00	0.383796
$484$	484.000	0.0454545
$485$	17810.0	1.66744
$486$	486.000	0.0453609
$487$	−19096.0	−1.77684	−0.888421	−	0.459029i	$-0.848197\pi$
$487$	−19096.0	−1.77684	−0.888421	+	0.459029i	$0.848197\pi$
$488$	−4240.00	−0.393311
$489$	2019.00	0.186712
$490$	−1274.00	−0.117456
$491$	−3705.00	−0.340538	−0.170269	−	0.985398i	$-0.554464\pi$
$491$	−3705.00	−0.340538	−0.170269	+	0.985398i	$0.554464\pi$
$492$	3504.00	0.321082
$493$	−1768.00	−0.161515
$494$	−2814.00	−0.256291
$495$	−1287.00	−0.116861
$496$	1408.00	0.127462
$497$	−4228.00	−0.381593
$498$	−864.000	−0.0777445
$499$	−3673.00	−0.329511	−0.164756	−	0.986334i	$-0.552684\pi$
$499$	−3673.00	−0.329511	−0.164756	+	0.986334i	$0.552684\pi$
$500$	4212.00	0.376733
$501$	5538.00	0.493851
$502$	8430.00	0.749501
$503$	−7326.00	−0.649404	−0.324702	−	0.945816i	$-0.605264\pi$
$503$	−7326.00	−0.649404	−0.324702	+	0.945816i	$0.605264\pi$
$504$	−504.000	−0.0445435
$505$	−25740.0	−2.26815
$506$	−4268.00	−0.374972
$507$	6876.00	0.602315
$508$	2432.00	0.212407
$509$	3470.00	0.302171	0.151086	−	0.988521i	$-0.451723\pi$
$509$	3470.00	0.302171	0.151086	+	0.988521i	$0.451723\pi$
$510$	−624.000	−0.0541788
$511$	−4921.00	−0.426012
$512$	512.000	0.0441942
$513$	567.000	0.0487986
$514$	−13890.0	−1.19195
$515$	8658.00	0.740810
$516$	−5496.00	−0.468891
$517$	2431.00	0.206799
$518$	4858.00	0.412062
$519$	−8448.00	−0.714501
$520$	6968.00	0.587629
$521$	6379.00	0.536409	0.268204	−	0.963362i	$-0.413570\pi$
$521$	6379.00	0.536409	0.268204	+	0.963362i	$0.413570\pi$
$522$	−3978.00	−0.333549
$523$	3671.00	0.306925	0.153462	−	0.988154i	$-0.450958\pi$
$523$	3671.00	0.306925	0.153462	+	0.988154i	$0.450958\pi$
$524$	−768.000	−0.0640272
$525$	−924.000	−0.0768127
$526$	−114.000	−0.00944988
$527$	704.000	0.0581911
$528$	528.000	0.0435194
$529$	25469.0	2.09329
$530$	16692.0	1.36803
$531$	2457.00	0.200800
$532$	−588.000	−0.0479192
$533$	−19564.0	−1.58989
$534$	4500.00	0.364670
$535$	−18707.0	−1.51173
$536$	4488.00	0.361664
$537$	8400.00	0.675022
$538$	5004.00	0.401000
$539$	539.000	0.0430730
$540$	−1404.00	−0.111886
$541$	9572.00	0.760688	0.380344	−	0.924845i	$-0.375806\pi$
$541$	9572.00	0.760688	0.380344	+	0.924845i	$0.375806\pi$
$542$	1130.00	0.0895529
$543$	6126.00	0.484147
$544$	256.000	0.0201763
$545$	12428.0	0.976802
$546$	2814.00	0.220564
$547$	8760.00	0.684736	0.342368	−	0.939566i	$-0.388771\pi$
$547$	8760.00	0.684736	0.342368	+	0.939566i	$0.388771\pi$
$548$	−4848.00	−0.377913
$549$	−4770.00	−0.370817
$550$	968.000	0.0750467
$551$	−4641.00	−0.358826
$552$	−4656.00	−0.359008
$553$	−3864.00	−0.297132
$554$	−14328.0	−1.09881
$555$	13533.0	1.03503
$556$	−7504.00	−0.572375
$557$	23651.0	1.79915	0.899574	−	0.436769i	$-0.143877\pi$
$557$	23651.0	1.79915	0.899574	+	0.436769i	$0.143877\pi$
$558$	1584.00	0.120172
$559$	30686.0	2.32179
$560$	1456.00	0.109870
$561$	264.000	0.0198683
$562$	−13970.0	−1.04856
$563$	5198.00	0.389111	0.194556	−	0.980891i	$-0.437674\pi$
$563$	5198.00	0.389111	0.194556	+	0.980891i	$0.437674\pi$
$564$	2652.00	0.197995
$565$	29926.0	2.22831
$566$	−12870.0	−0.955771
$567$	−567.000	−0.0419961
$568$	4832.00	0.356948
$569$	18306.0	1.34873	0.674365	−	0.738398i	$-0.264417\pi$
$569$	18306.0	1.34873	0.674365	+	0.738398i	$0.264417\pi$
$570$	−1638.00	−0.120365
$571$	−13676.0	−1.00232	−0.501158	−	0.865356i	$-0.667093\pi$
$571$	−13676.0	−1.00232	−0.501158	+	0.865356i	$0.667093\pi$
$572$	−2948.00	−0.215493
$573$	11976.0	0.873132
$574$	−4088.00	−0.297265
$575$	−8536.00	−0.619088
$576$	576.000	0.0416667
$577$	−6712.00	−0.484271	−0.242135	−	0.970242i	$-0.577848\pi$
$577$	−6712.00	−0.484271	−0.242135	+	0.970242i	$0.577848\pi$
$578$	−9698.00	−0.697896
$579$	4578.00	0.328593
$580$	11492.0	0.822723
$581$	1008.00	0.0719774
$582$	−8220.00	−0.585447
$583$	−7062.00	−0.501678
$584$	5624.00	0.398498
$585$	7839.00	0.554022
$586$	16800.0	1.18430
$587$	24891.0	1.75019	0.875095	−	0.483951i	$-0.160799\pi$
$587$	24891.0	1.75019	0.875095	+	0.483951i	$0.160799\pi$
$588$	588.000	0.0412393
$589$	1848.00	0.129279
$590$	−7098.00	−0.495288
$591$	1062.00	0.0739168
$592$	−5552.00	−0.385449
$593$	7896.00	0.546796	0.273398	−	0.961901i	$-0.411852\pi$
$593$	7896.00	0.546796	0.273398	+	0.961901i	$0.411852\pi$
$594$	594.000	0.0410305
$595$	728.000	0.0501598
$596$	−1060.00	−0.0728512
$597$	4422.00	0.303150
$598$	25996.0	1.77768
$599$	−4320.00	−0.294675	−0.147338	−	0.989086i	$-0.547070\pi$
$599$	−4320.00	−0.294675	−0.147338	+	0.989086i	$0.547070\pi$
$600$	1056.00	0.0718517
$601$	−9457.00	−0.641862	−0.320931	−	0.947103i	$-0.603996\pi$
$601$	−9457.00	−0.641862	−0.320931	+	0.947103i	$0.603996\pi$
$602$	6412.00	0.434109
$603$	5049.00	0.340980
$604$	−6912.00	−0.465638
$605$	−1573.00	−0.105705
$606$	11880.0	0.796356
$607$	28961.0	1.93656	0.968279	−	0.249871i	$-0.0803884\pi$
$607$	28961.0	1.93656	0.968279	+	0.249871i	$0.0803884\pi$
$608$	672.000	0.0448243
$609$	4641.00	0.308806
$610$	13780.0	0.914649
$611$	−14807.0	−0.980404
$612$	288.000	0.0190224
$613$	244.000	0.0160768	0.00803839	−	0.999968i	$-0.497441\pi$
$613$	244.000	0.0160768	0.00803839	+	0.999968i	$0.497441\pi$
$614$	−12728.0	−0.836580
$615$	−11388.0	−0.746680
$616$	−616.000	−0.0402911
$617$	−19062.0	−1.24377	−0.621886	−	0.783108i	$-0.713633\pi$
$617$	−19062.0	−1.24377	−0.621886	+	0.783108i	$0.713633\pi$
$618$	−3996.00	−0.260101
$619$	−18148.0	−1.17840	−0.589200	−	0.807987i	$-0.700557\pi$
$619$	−18148.0	−1.17840	−0.589200	+	0.807987i	$0.700557\pi$
$620$	−4576.00	−0.296414
$621$	−5238.00	−0.338476
$622$	−14912.0	−0.961281
$623$	−5250.00	−0.337619
$624$	−3216.00	−0.206319
$625$	−19189.0	−1.22810
$626$	20408.0	1.30298
$627$	693.000	0.0441400
$628$	−3536.00	−0.224684
$629$	−2776.00	−0.175972
$630$	1638.00	0.103586
$631$	8016.00	0.505724	0.252862	−	0.967502i	$-0.418628\pi$
$631$	8016.00	0.505724	0.252862	+	0.967502i	$0.418628\pi$
$632$	4416.00	0.277942
$633$	−1770.00	−0.111139
$634$	19332.0	1.21100
$635$	−7904.00	−0.493954
$636$	−7704.00	−0.480320
$637$	−3283.00	−0.204203
$638$	−4862.00	−0.301706
$639$	5436.00	0.336533
$640$	−1664.00	−0.102774
$641$	−19476.0	−1.20009	−0.600043	−	0.799967i	$-0.704850\pi$
$641$	−19476.0	−1.20009	−0.600043	+	0.799967i	$0.704850\pi$
$642$	8634.00	0.530774
$643$	5206.00	0.319292	0.159646	−	0.987174i	$-0.448965\pi$
$643$	5206.00	0.319292	0.159646	+	0.987174i	$0.448965\pi$
$644$	5432.00	0.332377
$645$	17862.0	1.09041
$646$	336.000	0.0204640
$647$	13227.0	0.803720	0.401860	−	0.915701i	$-0.368364\pi$
$647$	13227.0	0.803720	0.401860	+	0.915701i	$0.368364\pi$
$648$	648.000	0.0392837
$649$	3003.00	0.181630
$650$	−5896.00	−0.355785
$651$	−1848.00	−0.111258
$652$	2692.00	0.161698
$653$	−11768.0	−0.705233	−0.352617	−	0.935768i	$-0.614708\pi$
$653$	−11768.0	−0.705233	−0.352617	+	0.935768i	$0.614708\pi$
$654$	−5736.00	−0.342959
$655$	2496.00	0.148896
$656$	4672.00	0.278065
$657$	6327.00	0.375707
$658$	−3094.00	−0.183308
$659$	−8189.00	−0.484064	−0.242032	−	0.970268i	$-0.577814\pi$
$659$	−8189.00	−0.484064	−0.242032	+	0.970268i	$0.577814\pi$
$660$	−1716.00	−0.101205
$661$	−30562.0	−1.79837	−0.899186	−	0.437566i	$-0.855840\pi$
$661$	−30562.0	−1.79837	−0.899186	+	0.437566i	$0.855840\pi$
$662$	−22456.0	−1.31839
$663$	−1608.00	−0.0941924
$664$	−1152.00	−0.0673287
$665$	1911.00	0.111437
$666$	−6246.00	−0.363405
$667$	42874.0	2.48889
$668$	7384.00	0.427688
$669$	−8142.00	−0.470535
$670$	−14586.0	−0.841054
$671$	−5830.00	−0.335417
$672$	−672.000	−0.0385758
$673$	17626.0	1.00956	0.504779	−	0.863249i	$-0.331574\pi$
$673$	17626.0	1.00956	0.504779	+	0.863249i	$0.331574\pi$
$674$	4548.00	0.259915
$675$	1188.00	0.0677424
$676$	9168.00	0.521620
$677$	13610.0	0.772636	0.386318	−	0.922366i	$-0.373747\pi$
$677$	13610.0	0.772636	0.386318	+	0.922366i	$0.373747\pi$
$678$	−13812.0	−0.782370
$679$	9590.00	0.542018
$680$	−832.000	−0.0469202
$681$	−9978.00	−0.561465
$682$	1936.00	0.108700
$683$	−5328.00	−0.298492	−0.149246	−	0.988800i	$-0.547685\pi$
$683$	−5328.00	−0.298492	−0.149246	+	0.988800i	$0.547685\pi$
$684$	756.000	0.0422608
$685$	15756.0	0.878841
$686$	−686.000	−0.0381802
$687$	−3882.00	−0.215586
$688$	−7328.00	−0.406072
$689$	43014.0	2.37838
$690$	15132.0	0.834877
$691$	−6222.00	−0.342541	−0.171271	−	0.985224i	$-0.554787\pi$
$691$	−6222.00	−0.342541	−0.171271	+	0.985224i	$0.554787\pi$
$692$	−11264.0	−0.618776
$693$	−693.000	−0.0379869
$694$	−4016.00	−0.219662
$695$	24388.0	1.33106
$696$	−5304.00	−0.288861
$697$	2336.00	0.126947
$698$	3442.00	0.186650
$699$	10170.0	0.550307
$700$	−1232.00	−0.0665217
$701$	−2034.00	−0.109591	−0.0547954	−	0.998498i	$-0.517451\pi$
$701$	−2034.00	−0.109591	−0.0547954	+	0.998498i	$0.517451\pi$
$702$	−3618.00	−0.194519
$703$	−7287.00	−0.390945
$704$	704.000	0.0376889
$705$	−8619.00	−0.460440
$706$	−4110.00	−0.219096
$707$	−13860.0	−0.737283
$708$	3276.00	0.173898
$709$	18591.0	0.984767	0.492383	−	0.870378i	$-0.336126\pi$
$709$	18591.0	0.984767	0.492383	+	0.870378i	$0.336126\pi$
$710$	−15704.0	−0.830085
$711$	4968.00	0.262046
$712$	6000.00	0.315814
$713$	−17072.0	−0.896706
$714$	−336.000	−0.0176113
$715$	9581.00	0.501132
$716$	11200.0	0.584586
$717$	15627.0	0.813948
$718$	2608.00	0.135557
$719$	9619.00	0.498927	0.249463	−	0.968384i	$-0.419746\pi$
$719$	9619.00	0.498927	0.249463	+	0.968384i	$0.419746\pi$
$720$	−1872.00	−0.0968963
$721$	4662.00	0.240807
$722$	−12836.0	−0.661643
$723$	−7533.00	−0.387490
$724$	8168.00	0.419284
$725$	−9724.00	−0.498125
$726$	726.000	0.0371135
$727$	27818.0	1.41914	0.709568	−	0.704637i	$-0.248890\pi$
$727$	27818.0	1.41914	0.709568	+	0.704637i	$0.248890\pi$
$728$	3752.00	0.191014
$729$	729.000	0.0370370
$730$	−18278.0	−0.926711
$731$	−3664.00	−0.185387
$732$	−6360.00	−0.321137
$733$	10982.0	0.553383	0.276691	−	0.960959i	$-0.410762\pi$
$733$	10982.0	0.553383	0.276691	+	0.960959i	$0.410762\pi$
$734$	−3508.00	−0.176407
$735$	−1911.00	−0.0959024
$736$	−6208.00	−0.310910
$737$	6171.00	0.308428
$738$	5256.00	0.262163
$739$	1914.00	0.0952742	0.0476371	−	0.998865i	$-0.484831\pi$
$739$	1914.00	0.0952742	0.0476371	+	0.998865i	$0.484831\pi$
$740$	18044.0	0.896366
$741$	−4221.00	−0.209261
$742$	8988.00	0.444690
$743$	−2049.00	−0.101172	−0.0505858	−	0.998720i	$-0.516109\pi$
$743$	−2049.00	−0.101172	−0.0505858	+	0.998720i	$0.516109\pi$
$744$	2112.00	0.104072
$745$	3445.00	0.169416
$746$	22696.0	1.11389
$747$	−1296.00	−0.0634781
$748$	352.000	0.0172064
$749$	−10073.0	−0.491401
$750$	6318.00	0.307601
$751$	−7415.00	−0.360289	−0.180145	−	0.983640i	$-0.557657\pi$
$751$	−7415.00	−0.360289	−0.180145	+	0.983640i	$0.557657\pi$
$752$	3536.00	0.171469
$753$	12645.0	0.611965
$754$	29614.0	1.43034
$755$	22464.0	1.08285
$756$	−756.000	−0.0363696
$757$	8141.00	0.390871	0.195436	−	0.980717i	$-0.437388\pi$
$757$	8141.00	0.390871	0.195436	+	0.980717i	$0.437388\pi$
$758$	−17890.0	−0.857248
$759$	−6402.00	−0.306163
$760$	−2184.00	−0.104239
$761$	27564.0	1.31300	0.656501	−	0.754325i	$-0.272036\pi$
$761$	27564.0	1.31300	0.656501	+	0.754325i	$0.272036\pi$
$762$	3648.00	0.173429
$763$	6692.00	0.317519
$764$	15968.0	0.756154
$765$	−936.000	−0.0442368
$766$	−11120.0	−0.524519
$767$	−18291.0	−0.861082
$768$	768.000	0.0360844
$769$	−12095.0	−0.567174	−0.283587	−	0.958947i	$-0.591524\pi$
$769$	−12095.0	−0.567174	−0.283587	+	0.958947i	$0.591524\pi$
$770$	2002.00	0.0936975
$771$	−20835.0	−0.973222
$772$	6104.00	0.284570
$773$	−20925.0	−0.973635	−0.486818	−	0.873504i	$-0.661842\pi$
$773$	−20925.0	−0.973635	−0.486818	+	0.873504i	$0.661842\pi$
$774$	−8244.00	−0.382848
$775$	3872.00	0.179466
$776$	−10960.0	−0.507012
$777$	7287.00	0.336447
$778$	−24744.0	−1.14025
$779$	6132.00	0.282030
$780$	10452.0	0.479797
$781$	6644.00	0.304406
$782$	−3104.00	−0.141942
$783$	−5967.00	−0.272341
$784$	784.000	0.0357143
$785$	11492.0	0.522506
$786$	−1152.00	−0.0522780
$787$	−10789.0	−0.488674	−0.244337	−	0.969690i	$-0.578570\pi$
$787$	−10789.0	−0.488674	−0.244337	+	0.969690i	$0.578570\pi$
$788$	1416.00	0.0640138
$789$	−171.000	−0.00771579
$790$	−14352.0	−0.646356
$791$	16114.0	0.724334
$792$	792.000	0.0355335
$793$	35510.0	1.59016
$794$	−7968.00	−0.356138
$795$	25038.0	1.11699
$796$	5896.00	0.262535
$797$	−11011.0	−0.489372	−0.244686	−	0.969602i	$-0.578685\pi$
$797$	−11011.0	−0.489372	−0.244686	+	0.969602i	$0.578685\pi$
$798$	−882.000	−0.0391259
$799$	1768.00	0.0782820
$800$	1408.00	0.0622254
$801$	6750.00	0.297752
$802$	−828.000	−0.0364560
$803$	7733.00	0.339840
$804$	6732.00	0.295298
$805$	−17654.0	−0.772946
$806$	−11792.0	−0.515329
$807$	7506.00	0.327415
$808$	15840.0	0.689665
$809$	−35205.0	−1.52997	−0.764983	−	0.644051i	$-0.777252\pi$
$809$	−35205.0	−1.52997	−0.764983	+	0.644051i	$0.777252\pi$
$810$	−2106.00	−0.0913547
$811$	32951.0	1.42672	0.713358	−	0.700800i	$-0.247174\pi$
$811$	32951.0	1.42672	0.713358	+	0.700800i	$0.247174\pi$
$812$	6188.00	0.267434
$813$	1695.00	0.0731196
$814$	−7634.00	−0.328712
$815$	−8749.00	−0.376030
$816$	384.000	0.0164739
$817$	−9618.00	−0.411862
$818$	9068.00	0.387598
$819$	4221.00	0.180090
$820$	−15184.0	−0.646644
$821$	477.000	0.0202770	0.0101385	−	0.999949i	$-0.496773\pi$
$821$	477.000	0.0202770	0.0101385	+	0.999949i	$0.496773\pi$
$822$	−7272.00	−0.308565
$823$	20921.0	0.886100	0.443050	−	0.896497i	$-0.353896\pi$
$823$	20921.0	0.886100	0.443050	+	0.896497i	$0.353896\pi$
$824$	−5328.00	−0.225254
$825$	1452.00	0.0612753
$826$	−3822.00	−0.160998
$827$	−29965.0	−1.25996	−0.629979	−	0.776612i	$-0.716936\pi$
$827$	−29965.0	−1.25996	−0.629979	+	0.776612i	$0.716936\pi$
$828$	−6984.00	−0.293129
$829$	9232.00	0.386780	0.193390	−	0.981122i	$-0.438052\pi$
$829$	9232.00	0.386780	0.193390	+	0.981122i	$0.438052\pi$
$830$	3744.00	0.156574
$831$	−21492.0	−0.897171
$832$	−4288.00	−0.178677
$833$	392.000	0.0163049
$834$	−11256.0	−0.467342
$835$	−23998.0	−0.994593
$836$	924.000	0.0382263
$837$	2376.00	0.0981202
$838$	−10570.0	−0.435721
$839$	−27435.0	−1.12892	−0.564459	−	0.825461i	$-0.690915\pi$
$839$	−27435.0	−1.12892	−0.564459	+	0.825461i	$0.690915\pi$
$840$	2184.00	0.0897085
$841$	24452.0	1.00258
$842$	18038.0	0.738279
$843$	−20955.0	−0.856143
$844$	−2360.00	−0.0962495
$845$	−29796.0	−1.21303
$846$	3978.00	0.161662
$847$	−847.000	−0.0343604
$848$	−10272.0	−0.415969
$849$	−19305.0	−0.780384
$850$	704.000	0.0284082
$851$	67318.0	2.71167
$852$	7248.00	0.291446
$853$	7214.00	0.289569	0.144785	−	0.989463i	$-0.453751\pi$
$853$	7214.00	0.289569	0.144785	+	0.989463i	$0.453751\pi$
$854$	7420.00	0.297315
$855$	−2457.00	−0.0982779
$856$	11512.0	0.459664
$857$	28298.0	1.12794	0.563968	−	0.825797i	$-0.309274\pi$
$857$	28298.0	1.12794	0.563968	+	0.825797i	$0.309274\pi$
$858$	−4422.00	−0.175949
$859$	−31460.0	−1.24959	−0.624797	−	0.780787i	$-0.714818\pi$
$859$	−31460.0	−1.24959	−0.624797	+	0.780787i	$0.714818\pi$
$860$	23816.0	0.944324
$861$	−6132.00	−0.242715
$862$	8894.00	0.351428
$863$	−3014.00	−0.118885	−0.0594425	−	0.998232i	$-0.518932\pi$
$863$	−3014.00	−0.118885	−0.0594425	+	0.998232i	$0.518932\pi$
$864$	864.000	0.0340207
$865$	36608.0	1.43897
$866$	−7016.00	−0.275304
$867$	−14547.0	−0.569829
$868$	−2464.00	−0.0963521
$869$	6072.00	0.237029
$870$	17238.0	0.671750
$871$	−37587.0	−1.46221
$872$	−7648.00	−0.297011
$873$	−12330.0	−0.478015
$874$	−8148.00	−0.315343
$875$	−7371.00	−0.284783
$876$	8436.00	0.325372
$877$	25564.0	0.984304	0.492152	−	0.870509i	$-0.336210\pi$
$877$	25564.0	0.984304	0.492152	+	0.870509i	$0.336210\pi$
$878$	−26374.0	−1.01376
$879$	25200.0	0.966979
$880$	−2288.00	−0.0876460
$881$	−3945.00	−0.150863	−0.0754316	−	0.997151i	$-0.524033\pi$
$881$	−3945.00	−0.150863	−0.0754316	+	0.997151i	$0.524033\pi$
$882$	882.000	0.0336718
$883$	−2693.00	−0.102635	−0.0513175	−	0.998682i	$-0.516342\pi$
$883$	−2693.00	−0.102635	−0.0513175	+	0.998682i	$0.516342\pi$
$884$	−2144.00	−0.0815730
$885$	−10647.0	−0.404401
$886$	−16032.0	−0.607907
$887$	14500.0	0.548887	0.274443	−	0.961603i	$-0.411506\pi$
$887$	14500.0	0.548887	0.274443	+	0.961603i	$0.411506\pi$
$888$	−8328.00	−0.314718
$889$	−4256.00	−0.160564
$890$	−19500.0	−0.734429
$891$	891.000	0.0335013
$892$	−10856.0	−0.407495
$893$	4641.00	0.173914
$894$	−1590.00	−0.0594827
$895$	−36400.0	−1.35946
$896$	−896.000	−0.0334077
$897$	38994.0	1.45147
$898$	−11448.0	−0.425417
$899$	−19448.0	−0.721498
$900$	1584.00	0.0586667
$901$	−5136.00	−0.189906
$902$	6424.00	0.237135
$903$	9618.00	0.354449
$904$	−18416.0	−0.677552
$905$	−26546.0	−0.975049
$906$	−10368.0	−0.380192
$907$	18908.0	0.692205	0.346102	−	0.938197i	$-0.387505\pi$
$907$	18908.0	0.692205	0.346102	+	0.938197i	$0.387505\pi$
$908$	−13304.0	−0.486243
$909$	17820.0	0.650222
$910$	−12194.0	−0.444206
$911$	−42894.0	−1.55998	−0.779990	−	0.625792i	$-0.784776\pi$
$911$	−42894.0	−1.55998	−0.779990	+	0.625792i	$0.784776\pi$
$912$	1008.00	0.0365989
$913$	−1584.00	−0.0574181
$914$	4612.00	0.166905
$915$	20670.0	0.746808
$916$	−5176.00	−0.186703
$917$	1344.00	0.0484000
$918$	432.000	0.0155317
$919$	−16050.0	−0.576105	−0.288053	−	0.957615i	$-0.593008\pi$
$919$	−16050.0	−0.576105	−0.288053	+	0.957615i	$0.593008\pi$
$920$	20176.0	0.723025
$921$	−19092.0	−0.683065
$922$	−32136.0	−1.14788
$923$	−40468.0	−1.44314
$924$	−924.000	−0.0328976
$925$	−15268.0	−0.542712
$926$	−32374.0	−1.14889
$927$	−5994.00	−0.212372
$928$	−7072.00	−0.250161
$929$	−17595.0	−0.621392	−0.310696	−	0.950509i	$-0.600562\pi$
$929$	−17595.0	−0.621392	−0.310696	+	0.950509i	$0.600562\pi$
$930$	−6864.00	−0.242021
$931$	1029.00	0.0362235
$932$	13560.0	0.476580
$933$	−22368.0	−0.784883
$934$	−27006.0	−0.946106
$935$	−1144.00	−0.0400137
$936$	−4824.00	−0.168459
$937$	55242.0	1.92602	0.963008	−	0.269472i	$-0.0868491\pi$
$937$	55242.0	1.92602	0.963008	+	0.269472i	$0.0868491\pi$
$938$	−7854.00	−0.273393
$939$	30612.0	1.06388
$940$	−11492.0	−0.398753
$941$	−27020.0	−0.936054	−0.468027	−	0.883714i	$-0.655035\pi$
$941$	−27020.0	−0.936054	−0.468027	+	0.883714i	$0.655035\pi$
$942$	−5304.00	−0.183454
$943$	−56648.0	−1.95622
$944$	4368.00	0.150600
$945$	2457.00	0.0845780
$946$	−10076.0	−0.346299
$947$	−6874.00	−0.235876	−0.117938	−	0.993021i	$-0.537628\pi$
$947$	−6874.00	−0.235876	−0.117938	+	0.993021i	$0.537628\pi$
$948$	6624.00	0.226938
$949$	−47101.0	−1.61113
$950$	1848.00	0.0631127
$951$	28998.0	0.988775
$952$	−448.000	−0.0152519
$953$	5615.00	0.190858	0.0954290	−	0.995436i	$-0.469578\pi$
$953$	5615.00	0.190858	0.0954290	+	0.995436i	$0.469578\pi$
$954$	−11556.0	−0.392180
$955$	−51896.0	−1.75845
$956$	20836.0	0.704900
$957$	−7293.00	−0.246342
$958$	−3176.00	−0.107111
$959$	8484.00	0.285675
$960$	−2496.00	−0.0839146
$961$	−22047.0	−0.740056
$962$	46498.0	1.55837
$963$	12951.0	0.433375
$964$	−10044.0	−0.335576
$965$	−19838.0	−0.661770
$966$	8148.00	0.271385
$967$	10562.0	0.351242	0.175621	−	0.984458i	$-0.443807\pi$
$967$	10562.0	0.351242	0.175621	+	0.984458i	$0.443807\pi$
$968$	968.000	0.0321412
$969$	504.000	0.0167088
$970$	35620.0	1.17906
$971$	−31669.0	−1.04666	−0.523330	−	0.852130i	$-0.675310\pi$
$971$	−31669.0	−1.04666	−0.523330	+	0.852130i	$0.675310\pi$
$972$	972.000	0.0320750
$973$	13132.0	0.432675
$974$	−38192.0	−1.25642
$975$	−8844.00	−0.290497
$976$	−8480.00	−0.278113
$977$	24134.0	0.790292	0.395146	−	0.918618i	$-0.370694\pi$
$977$	24134.0	0.790292	0.395146	+	0.918618i	$0.370694\pi$
$978$	4038.00	0.132026
$979$	8250.00	0.269327
$980$	−2548.00	−0.0830540
$981$	−8604.00	−0.280025
$982$	−7410.00	−0.240797
$983$	−2868.00	−0.0930570	−0.0465285	−	0.998917i	$-0.514816\pi$
$983$	−2868.00	−0.0930570	−0.0465285	+	0.998917i	$0.514816\pi$
$984$	7008.00	0.227040
$985$	−4602.00	−0.148865
$986$	−3536.00	−0.114208
$987$	−4641.00	−0.149670
$988$	−5628.00	−0.181225
$989$	88852.0	2.85675
$990$	−2574.00	−0.0826334
$991$	−1271.00	−0.0407413	−0.0203707	−	0.999792i	$-0.506485\pi$
$991$	−1271.00	−0.0407413	−0.0203707	+	0.999792i	$0.506485\pi$
$992$	2816.00	0.0901291
$993$	−33684.0	−1.07646
$994$	−8456.00	−0.269827
$995$	−19162.0	−0.610529
$996$	−1728.00	−0.0549737
$997$	8706.00	0.276551	0.138276	−	0.990394i	$-0.455844\pi$
$997$	8706.00	0.276551	0.138276	+	0.990394i	$0.455844\pi$
$998$	−7346.00	−0.233000
$999$	−9369.00	−0.296719

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	462.4.a.i.1.1	✓	1
3.2	odd	2		1386.4.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.4.a.i.1.1	✓	1	1.1	even	1	trivial
1386.4.a.f.1.1		1	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	462.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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